**0.(4) = 0.444444444444444444444444444444444444444**....,

is a number where the fourths go on forever, then

**0.(4)1 = 0.44444444444444444444444444444444444444....1**,

is a number where the fourths go on forever and, afterwards, there’s a one.

Imagine writing the number

**0.(4)1**into a grid. You fill the first line with fourths and then write "1" into the second line like this

Obviously,

**0.(4)**equals

**0.(44)**or

**0.(444)**(=one line of 4s) but not

**0.(4)(4)**(=two lines of 4s). For complex decimals with repetitions of repetitions, like

**0.(1(2(3)))((4))56**, you need N-dimensional grid, but the principle is analogous.

Ben asked the question whether Kaufman decimals can be totally ordered. Sure - just look for the first digit that differs. To do this precisely, one needs ordinal indices - Mariano Chouza provides a proof on his blog. It has been a long time since I have seen a set-theory stuff like this. I more or less guess than really understand it but the main idea is easy to comprehend.

So, how to compare Kaufman decimals on a computer? Jeff Kaufman upload some Python code to Github that is not really working (0.(81)>0.89 and other issues). I cloned his project and here is my own attempt:

- The first difference matters. So I implemented "split" function that returns the first omega^k digits (k=0 one digit, k=1 a line, k=2 plane, ...) and the rest of the number
- I start from the beginning, a comparison of
**0.(4715)**and**0.471548**goes like this: **0.(4715) =****0.4(7154)**has the same first digit as**0.471548,**cut it off from both**0.(7154) =****0.7(1547)****0.71548,**cut it off**0.(1547) =****0.1(5471)****0.1548,**cut it off**0.(5471) =****0.5(4715)****0.548,**cut it off**0.(4715) =****0.4(7154)****0.48,**cut it off**0.(7154) =****0.7(7154)****0.8, so****0.(4715)**<**0.471548**- The situation might be simply more complicated inside the repetition, say
**0.47(4747)8**and**0.(474747)8**, then it goes like this: **0.(474747)8 =****0.4(**has the same first digit as**747474**)8cut it off from both**0.47(4747)8**,**0.(747474)8**=**0.7(474747)8****0.**cut it off,**7(4747)8****0.(474747)****0.****(4747)**are both one line numbers (same order of infinity), compare them- the first digits equal, cut it off
- the second digits equal, cut it off
**hey,**I was in this comparison before, so**0.(474747)****0.****(4747)****0.8 = 0.8, hence****0.47(4747)8**=**0.(474747)8**

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